Embedded newform 201.3.g.b.29.14

• Label: 201.3.g.b.29.14
• Level: $201$
• Weight: $3$
• Character: 201.29
• Analytic conductor: $5.477$
• Analytic rank: $0$
• Dimension: $84$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	201.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.47685331364\)
Analytic rank:	\(0\)
Dimension:	\(84\)
Relative dimension:	\(42\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			29.14
Character	\(\chi\)	\(=\)	201.29
Dual form			201.3.g.b.104.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34106 - 0.774260i) q^{2} +(-2.89057 - 0.802859i) q^{3} +(-0.801042 - 1.38745i) q^{4} +1.39955i q^{5} +(3.25481 + 3.31474i) q^{6} +(3.37843 + 5.85161i) q^{7} +8.67494i q^{8} +(7.71083 + 4.64145i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 6 q^{3} + 82 q^{4} + 3 q^{6} - 10 q^{7} + 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34106	−	0.774260i	−0.670529	−	0.387130i	0.125748	−	0.992062i	\(-0.459867\pi\)
							−0.796277	+	0.604932i	\(0.793200\pi\)
\(3\)	−2.89057	−	0.802859i	−0.963525	−	0.267620i
							\(5\)			1.39955i			0.279910i	0.990158	+	0.139955i	\(0.0446959\pi\)
−0.990158	+	0.139955i	\(0.955304\pi\)
\(7\)	3.37843	+	5.85161i	0.482632	+	0.835944i	0.999801	−	0.0199397i	\(-0.00634743\pi\)
							−0.517169	+	0.855883i	\(0.673014\pi\)
\(11\)	−5.65757	+	3.26640i	−0.514324	+	0.296945i	−0.734609	−	0.678490i	\(-0.762635\pi\)
							0.220285	+	0.975436i	\(0.429301\pi\)
\(13\)	10.2581	−	17.7676i	0.789087	−	1.36674i	−0.137440	−	0.990510i	\(-0.543887\pi\)
							0.926527	−	0.376229i	\(-0.122779\pi\)
\(17\)	−17.0786	−	9.86033i	−1.00462	−	0.580019i	−0.0950100	−	0.995476i	\(-0.530288\pi\)
							−0.909613	+	0.415457i	\(0.863622\pi\)
\(19\)	7.18817	−	12.4503i	0.378325	−	0.655278i	−0.612494	−	0.790475i	\(-0.709834\pi\)
							0.990819	+	0.135197i	\(0.0431669\pi\)
\(23\)	28.7870	+	16.6202i	1.25161	+	0.722616i	0.971428	−	0.237332i	\(-0.0762731\pi\)
							0.280178	+	0.959948i	\(0.409606\pi\)
\(29\)	14.9685	−	8.64206i	0.516155	−	0.298002i	−0.219205	−	0.975679i	\(-0.570346\pi\)
							0.735360	+	0.677677i	\(0.237013\pi\)
\(31\)	1.53525	+	2.65913i	0.0495242	+	0.0857784i	0.889725	−	0.456497i	\(-0.150896\pi\)
							−0.840201	+	0.542276i	\(0.817563\pi\)
\(37\)	1.29291	−	2.23938i	0.0349434	−	0.0605238i	−0.848025	−	0.529957i	\(-0.822208\pi\)
							0.882968	+	0.469433i	\(0.155542\pi\)
\(41\)	−31.0286	+	17.9144i	−0.756794	+	0.436935i	−0.828144	−	0.560516i	\(-0.810603\pi\)
							0.0713493	+	0.997451i	\(0.477270\pi\)
\(43\)	16.5986			0.386015			0.193007	−	0.981197i	\(-0.438176\pi\)
							0.193007	+	0.981197i	\(0.438176\pi\)
\(47\)	79.2154	−	45.7350i	1.68543	−	0.973086i	0.727496	−	0.686112i	\(-0.240684\pi\)
							0.957938	−	0.286974i	\(-0.0926495\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.g.b.29.14	✓	84
3.2	odd	2	inner	201.3.g.b.29.29	yes	84
67.37	even	3	inner	201.3.g.b.104.29	yes	84
201.104	odd	6	inner	201.3.g.b.104.14	yes	84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.g.b.29.14	✓	84	1.1	even	1	trivial
201.3.g.b.29.29	yes	84	3.2	odd	2	inner
201.3.g.b.104.14	yes	84	201.104	odd	6	inner
201.3.g.b.104.29	yes	84	67.37	even	3	inner